Existence of solutions for quasilinear parabolic equations

try to enter to this link which is an article about [EXISTENCE OF SOLUTIONS FOR QUASILINEAR PARABOLIC
EQUATIONS WITH NONLOCAL BOUNDARY CONDITIONS], I have question in the page 3 section 3:
the expression:
[We need a priori estimates that permit us to extend the solution to the whole
domain [0, T].] what does it mean this expression mathematicaly?????